# Symmetric and Skew Symmetric Matrices

### Symmetric Matrix:

A square matrix $A$ is symmetric if it is equal to its transpose: $A={A}^{T}$

• Properties:
• The elements along the main diagonal remain unchanged when transposed.
• ${a}_{ij}={a}_{ji}$ for all $i$ and $j$.
• It is always a square matrix (number of rows = number of columns).

### Example of Symmetric Matrix:

$A=\left[\begin{array}{ccc}2& 5& 7\\ 5& 3& 9\\ 7& 9& 1\end{array}\right]$

In this case, $A={A}^{T}$, which makes it a symmetric matrix.

### Skew-Symmetric Matrix:

A square matrix $A$ is skew-symmetric if it is equal to the negative of its transpose: $A=-{A}^{T}$

• Properties:
• The elements along the main diagonal are zero.
• ${a}_{ij}=-{a}_{ji}$ for all $i$ and $j$.
• It is always a square matrix (number of rows = number of columns).

### Example of Skew-Symmetric Matrix:

$A=\left[\begin{array}{ccc}0& 7& -3\\ -7& 0& 2\\ 3& -2& 0\end{array}\right]$

In this case, $A=-{A}^{T}$, which makes it a skew-symmetric matrix.

### Properties of Symmetric and Skew-Symmetric Matrices:

1. Sum of Symmetric Matrices: The sum of two symmetric matrices is also symmetric. $A+B=C$where $A$, $B$, and $C$ are symmetric matrices.

2. Sum of Skew-Symmetric Matrices: The sum of two skew-symmetric matrices is also skew-symmetric. $A+B=C$ where $A$, $B$, and $C$ are skew-symmetric matrices.

3. Product with Scalar: Both symmetric and skew-symmetric matrices preserve their property when multiplied by a scalar.